If you are a parent, you know exactly what it is like to argue with someone who is unreasonable and irrational. Many attempts at logic and reason end with the parent coming down to the level of the child—basing arguments on emotion usually in the form of a tasty bribe or smacked bottom, depending on what the circumstances call for. Unfortunately, many people carry these success-repelling traits with them into adulthood. This makes communication, cooperation, and prosperity a real challenge.
As you might have guessed, those who are unreasonable and irrational are either incapable or unwilling to accept that their arguments are fallacious, if in fact, they are. In these cases, you can come down to their level, appeal to their emotions, and exploit their cognitive biases—but this takes some manipulative talent, and I would argue that it is not very ethical. You can simply give up and refuse to argue any further, which I have done at times. Or, if possible, you can show how their arguments and beliefs are inconsistent with other beliefs they hold. This is my preferred strategy because it is not patronizing, nor does it reflect my frustration.